8.8 Continuous equations and regularityD5976AE10288

Proof. Smooth solutions exist by 8.6.1. At a maximum of $w_1-w_2$, monotonicity of the smooth Hessian operator gives $e^{aw_1+H_1}\leq e^{aw_2+H_2}$. Hence $a(w_1-w_2)\leq H_2-H_1$ there. Exchanging the functions gives the stated norm estimate for smooth data. Thus a uniformly convergent sequence of smooth data gives a uniformly Cauchy sequence of solutions. Its limit is admissible by the linear distributional inequalities, and 8.7.1 passes the equation to the limit. The bounds follow from those for the smooth solutions. Taking limits in the smooth norm estimate proves it for continuous data. Finally, 8.7.9 in both directions gives uniqueness among continuous admissible solutions. $\square$

Proof. Choose smooth admissible $v_ j\to \varphi $ uniformly, by 8.6.2. Their values lie in a fixed compact interval on which $f$ is smooth. Put $h(x,t)=-t+f(x,t)$ and solve

The preceding stability proposition shows that $u_ j$ converges uniformly to a continuous admissible $u$ satisfying

8.7.9, applied in both directions, gives $u=\varphi $.

The zero-order bounds for $u_ j$ are uniform. Fix $M$ bounding them and choose $B>\| \varphi \| _\infty +1$. Extend $h$ smoothly away from a neighborhood of the range of $\varphi $ to $X\times [-B-1,B+1]$; this does not change the equations for large $j$. The tolerance $\varepsilon $ in 8.4.6 depends on these bounds and on $h$, without derivatives of the smooth comparison function. Choose a fixed smooth $\widetilde\varphi $ with $\| \widetilde\varphi -\varphi \| _\infty <\varepsilon /2$ and $\| \widetilde\varphi \| _\infty \leq B$. For large $j$, $\| v_ j-\widetilde\varphi \| _\infty <\varepsilon $. 8.4.7 then gives uniform gradient and complex-Hessian bounds for $u_ j$, independent of derivatives of $v_ j$. 8.5.1 yields $\varphi \in W^{2,p}$ for every finite $p$, and $\varphi \in C^{1,\alpha }$ for every $\alpha <1$.

8.5.4 makes $\varphi $ a continuous viscosity solution of the concave uniformly elliptic extension, with right-hand side $e^{f(x,\varphi )/m}$. This right-hand side is Hölder. The analytic estimate in section 8.5 therefore gives $\varphi \in C^{2,\beta }$, for some $\beta >0$. The operator agrees with the original Hessian root on the attained compact cone set; the now classical equation thus has smooth dependence on $x,\varphi $ and its complex Hessian, and is uniformly elliptic there. Difference quotients and interior Schauder estimates give $C^{3,\beta }$; iteration proves smoothness. The estimate applies directly to the viscosity limit. $\square$

Proof. Choose $a\geq L$ and let $\mathcal C=\{ v\in C(X):A\leq v\leq B\} $, a complete subset of $C(X)$ in the uniform norm. For $v\in \mathcal C$ put $H_ v(x)=f(x,v(x))-av(x)$ and let $T(v)$ be the continuous proper solution with exponent $aT(v)+H_ v$, furnished by 8.8.1. The function $f(x,t)-at$ is nonincreasing in $t$, so

The zero-order bounds in that proposition give $A\leq T(v)\leq B$. The mean-value theorem also gives

Since $0\leq 1-\kappa /a<1$, the contraction theorem gives a unique fixed point $w\in \mathcal C$. It is admissible because it equals $T(w)$, and its equation is exactly the stated equation. Every continuous admissible solution in this interval is a fixed point, by uniqueness of the continuous proper equation with its prescribed $H_ w$. This proves uniqueness in the stated class. 8.8.2 makes $w$ smooth. Its form lies in the closed $m$-cone and has positive $m$th symmetric polynomial; Maclaurin's inequalities then put it in the open cone. Thus it is strictly admissible. $\square$